ghost effect by curvature in planar couette...

Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XX

GHOST EFFECT BY CURVATURE IN PLANAR COUETTEFLOW

L. Arkeryd

Chalmers Institute of Technology, Gothenburg, Sweden.

R. Esposito

Dipartimento di Matematica, Universita di L’Aquila, Coppito, 67100 AQ, Italy

R. Marra

Dipartimento di Fisica and Unita INFN, Universita di Roma Tor Vergata, 00133 Roma, Italy

A. Nouri

LATP, Universite d’Aix-Marseille I, Marseille, France.

(Communicated by the associate editor name)

To the memory of Carlo Cercignani

Abstract. We study a rarefied gas, described by the Boltzmann equation,

between two coaxial rotating cylinders in the small Knudsen number regime.When the radius of the inner cylinder is suitably sent to infinity, the limit-

ing evolution is expected to converge to a modified Couette flow which keeps

memory of the vanishing curvature of the cylinders (ghost effect [17]). In the1-d stationary case we prove the existence of a positive isolated L2-solution

to the Boltzmann equation and its convergence. This is obtained by meansof a truncated bulk-boundary layer expansion which requires the study of a

new Milne problem, and an estimate of the remainder based on a generalized

spectral inequality.

1. Introduction. It is well known (see for example [9, 17, 14] and references quotedtherein) that the asymptotic behavior of the solutions to the Boltzmann equation, inthe limit of small Knudsen numbers, is well approximated by the compressible Eulerequation for a perfect gas, while the viscosity and heat conducting effects are seenas first order corrections in the Knudsen number. To get finite size viscosity effectis more delicate because of the von Karman relation [20] between the Reynolds,Knudsen and Mach numbers, Re, Kn, Ma:

Ma ∼ ReKn.When Kn is small, either Re−1 or Ma have to be small. Therefore, finite viscosityeffects are attained only if one assumes that the Mach number is of the same orderas the Knudsen number.

With the extra assumption that density and temperature profiles differ fromconstants at most for terms of the order of the Knudsen number, it is then possibleto show that the asymptotic behavior of the solutions to the Boltzmann equation

2000 Mathematics Subject Classification. Primary: 82B40, 82C26; Secondary 76P05.Key words and phrases. Boltzmann equation, Hydrodynamical limit, Ghost effect, Couette

flow.

1

2 L. ARKERYD, R. ESPOSITO, R. MARRA, A. NOURI

are well approximated by the incompressible Navier-Stokes equations (INS), in thesense that the average velocity field, rescaled by the Mach number, converges to asolution u to INS. Moreover, the first order correction to the temperature profileconverges to a solution to the heat equation with a convective term due to therescaled velocity field u. Such results have been proved in several papers. Anoverview is provided in [14], to which we refer for a partial list of references onthe subject. We also stress that the asymptotic behavior of the solutions to thecompressible Navier-Stokes-Fourier equations, in the low Mach number limit andwith the same assumption on density and temperature, is the same.

When the density and temperature do not satisfy the above mentioned assump-tions, the Boltzmann equation deviates from the compressible Navier-Stokes-Fourierequations. Such a discrepancy, called ghost effect [17], is the issue we want to addressin this paper.

The name is suggested by the fact that a small velocity field produces finite sizemodifications of the usual heat equation. This fact is confirmed both numericallyand experimentally and devices as a vacuum pump have been designed startingfrom this effect. There has been a big theoretical, numerical and experimental workon this and for the details we refer to [17, 18]. We remark that the time dependentequations on the torus, in the diffusive space-time scaling were written in [11] andanalyzed in [7], where it is observed that similar equations had been previouslyconsidered in some special cases [15]. Very little is known from the mathematicalpoint of view, and the only rigorous result we are aware of is [8]. By using thetechniques illustrated in the present paper, it is possible to deal with the timedependent problem in the torus, but that will be the subject of future works.

In this paper instead, we want to study a different type of ghost effect, pointedout in [19, 17] as ghost effect by curvature. It consists in the following: if one looksat the Couette flow between two coaxial rotating cylinders in the limit when theradius of the inner cylinder goes to infinity, one expects to obtain, as asymptoticbehavior, the standard planar Couette flow. The analysis based on the Boltzmannequation does not confirm this. Indeed, an extra term appears in the limitingequations, which is reminiscent of the original structure of the problem. Hence,an infinitesimal curvature produces a finite size discrepancy, the ghost effect bycurvature. The extra term gives rise to a bifurcation of the laminar stationarysolution, which is absent in the standard Couette flow.

To be more specific, we look at the behavior of a rarefied gas between two coaxialcylinders of radius L and L+1, described by the Boltzmann equation in the diffusivespace-time scaling. In cylindrical coordinates (r, θ, z) ∈ (L,L+ 1)× [0, 2π)×R it iswritten as

ε∂F

∂t+ vr

∂F

∂r+vθr

∂F

∂θ+ vz

∂F

∂z+vθr

(vθ∂F

∂vr− vr

∂F

∂vθ

)=

1εQ(F, F ), (1.1)

where the positive and normalized function F (r, θ, z, vr, vθ, vz, t) is the probabilitydensity of particles in cylindrical coordinates and we have denoted by ε the Knudsennumber. (vr, vθ, vz) are the components of the velocity in the local basis associatedto the cylindrical coordinates and Q(f, g) is the Boltzmann collision integral forhard spheres:

Q(f, g)(v) =12

∫R3dv∗

∫S2

dnB(n, v − v∗)f ′∗g′ + f ′g′∗ − f∗g − g∗f

, (1.2)

GHOST EFFECT BY CURVATURE 3

with f ′, f ′∗, f, f∗ standing for f(v′), f(v′∗), f(v), f(v∗) respectively, S2 = n ∈ R3 |n2 =1, B is the differential cross section 2B(n, V ) = |V · n| corresponding to hardspheres, and v, v∗ and v′,v′∗ are post-collisional and precollisional velocities in anelastic collision:

v′ = v − n(v − v∗) · n, v′∗ = v∗ + n(v − v∗) · n. (1.3)

The use of cylindrical coordinates produces a force-like term depending on thevelocity, which will be referred below as centrifugal force.

We will look at the above equation in the planar limit, where one takes the radiusof the inner cylinder L to infinity. A convenient change of variables is the following:

r = L+y + π

2π; Lθ = −x; vr = vy; vθ = −vx,

while the z variable is unchanged. For simplicity, we assume the distribution Finvariant under rotations around the axis of the cylinders, so we drop the dependenceon θ. Moreover, we scale 1/L proportionally to the inverse of the square of theKnudsen number:

1L

=ε2

c2, (1.4)

with a constant c, related to the curvature, which will be specified below. Withthese assumptions the equation becomes

ε∂F

∂t+ vy

∂F

∂y+ vz

∂F

∂z+ε2

c2σ(y)vx

(vx∂F

∂vy− vy

∂F

∂vx

)=

1εQ(F, F ), (1.5)

with

σ(y) =2π

2π + ε2

c2 (y + π). (1.6)

The variable y varies between −π, corresponding to the inner cylinder, and π cor-responding to the outer cylinder.

The boundary conditions on the two cylinders are assumed to be given by thediffuse reflection condition, meaning that the distribution of the incoming particlesis Maxwellian:

F (−π, z, v, t) = α−(F )M−, vy > 0,(1.7)

F (π, z, v, t) = α+(F )M+, vy < 0.

We use the following notation: for ρ > 0, T > 0 and u ∈ R3

M(ρ, T, u; v) =ρ

(2πT )3/2e−|v − u|2

2T (1.8)

is the Maxwellian with density ρ, temperature T and mean velocity u. In this paperwe consider the two cylinders at the same temperature T = 1 and rotating withvelocities U− and U+. Therefore

M± = M(√

2π, 1, (U±, 0, 0); v), (1.9)

where the density has been fixed so that the normalization condition∫vy≷0

dv|vy|M± = 1 (1.10)


is satisfied. The constants α±, depending on the outgoing flow at the boundaries,are determined by the condition of vanishing net flow in the radial direction:∫

R3dvvyF (y, z, vx, vy, vz, t) = 0, for y = −π, π. (1.11)

By using (1.7) and (1.10), one immediately gets

α−(F ) = −∫vy<0

dvvyF (−π, z, vx, vy, vz, t),

(1.12)

α+(F ) =∫vy>0

dvvyF (π, z, vx, vy, vz, t).

We need to introduce a low Mach number assumption. Due to the particulargeometry we consider here, we do not need that the full velocity field is small.Indeed the tangential component can be of order 1, while we need the radial andaxial components to be of the order of the Knudsen number ε. We will use thenotation v to denote the couple (vy, vz). Correspondingly, ∇ = (∂y, ∂z). The Machnumber assumption is therefore:

u :=∫

R3dvvF = O(ε). (1.13)

The tangential component of the velocity, denoted by U is O(1) with respect to theKnudsen number ε. However, we will also need some smallness assumption on it.Therefore, we introduce another parameter δ, measuring the size of U :

U :=∫

R3dvvxF = O(δ). (1.14)

The fact that δ, although small, will be chosen much larger than ε, is responsibleof the arising of a ghost effect. In principle δ might be completely independent ofε, but, for technical reasons, we will assume, in the estimate of the remainder, aspecific relation between δ and ε. Therefore, from now on, we replace U with δUwith U = O(1) both in ε and δ. In order to get equations without singular termswhen δ → 0, we will also assume the constant c appearing in the definition of σ(1.6) of order δ:

c = δC (1.15)for some other constant C also of order 1.

As usual we will look for a solution to (1.5) in terms of a truncated expansion in ε.The collision term forces the lowest order of the expansion to be a local Maxwellian.In order to fulfil the low Mach number assumptions (1.13), (1.14) and the boundaryconditions (1.7), the lowest order has to be of the form

Mδ = M(1 + δr, 1 + δτ, (δU, 0, 0); v) =1 + δr

(2π(1 + δτ))3/2exp

[− v2

2(1 + δτ)

], (1.16)

wherev = v − (δU, 0, 0) = (vx − δU, v). (1.17)

Note that the functions r and τ , representing corrections of order δ to the densityand temperature, vanish at the boundary because we are restricting ourselves tothe case when the two cylinders are at the same temperature. On the other handgradients of U of order δ warm up the fluid in the bulk and may produce variationsof the temperature and density of order δ2.


The solution is sought for in the form

F = Mδ + Φ + εR, (1.18)

where

Φ =N∑n=1

εnFn (1.19)

for a suitable choice of N , and R is a remainder. In the next section we will give theprocedure to compute the functions Fn’s, which will be based on a kind of Hilbertexpansion for the bulk parts Bn of Fn and a boundary layer expansion in order torestore the boundary conditions violated by the bulk terms. The computation ofthe bulk terms requires the solution of a rather complex system of equations for thehydrodynamical fields U , u, r, τ depending on δ. They are

∇[r + τ ] + δ∇[rτ ] = 0,

∇ · u+ δ(∂tr + ∇ · (ur)) = 0,

(1 + δr)(∂tU + u · ∇U

)= η0∆U + ∇ · (ηδ∇U),

(1 + δr)(∂tu+ u · ∇u

)+ ∇P2 −

1C2

ρU2ey = η0∆u (1.20)

+∇ ·(ηδ∇u+

δ2

P

[σ1∇τ ⊗ ∇τ + σ2∇U ⊗ ∇U

]),

32

(1 + δr)∂tτ +52

(1 + δr)u · ∇τ = κ0∆τ + ∇(κδ∇τ) + δη|∇U |2.

Here η, κ, σ1 and σ2 are suitable transport coefficients depending on 1 + δτ . Wehave set η = η0 + ηδ, κ = κ0 + κδ, with η0 and κ0 the values corresponding toδ = 0 and ηδ, κδ the differences. ey is the unit vector in the direction y, P is definedin (2.18) and P2 is an unknown pressure related to the almost incompressibilitycondition given by the second of the equations (1.20).

When δ goes to 0 the equations take a rather simpler form:

∇ · u = 0,

∂tU + u · ∇U = η0∆U, (1.21)

∂tu+ u · ∇u+ ∇P2 −1C2

ρU2ey = η0∆u,

∇[r + τ ] = 0,32∂tτ +

52u · ∇τ = κ0∆τ. (1.22)

The equations have to be completed with Cauchy initial data and the time inde-pendent boundary conditions

u(±π, z) = 0, U(±π, z) = U±, τ(±π, z) = 0.

The first of the equations (1.22) is a Boussinesq condition ensuring the constancyof the pressure to the first order in δ, while the second is just the heat equation witha convective term. Note that the equations (1.21) are decoupled from the (1.22)and can be solved independently.

The equations (1.21) are the equations for the planar Couette system with anextra term in the equation for u, representing the curvature ghost effect. The linearanalysis for them [17, 19] shows the presence of a bifurcation controlled by the


parameters C and U±, with a stationary laminar solution losing its stability andbifurcating into two stable non laminar solutions.

This paper is devoted to the analysis of the 1-d stationary laminar solution to(1.5). The two dimensional case where bifurcation arises, will be presented in aforthcoming paper.

In Section 2 will be given a perturbative analysis in δ of the system (2.31) in orderto control the difference between its solutions and those to the equations (2.32).

Due to the presence of the centrifugal force, the boundary layer expansion, as in[13, 1, 2], has to include that force. This requires the solution of a Milne problemwith a force, which has been given in [10] in the presence of a potential force.The present force is different because it is not potential and depends on velocities.Therefore the arguments in [10] require several modifications which are presentedin Section 3. Finally, in Section 4 we estimate the remainder. The key ingredientto do this is a generalized spectral inequality for a perturbed linearized Boltzmannoperator, already used in [2] in the context of the Benard problem. The presence ofthe O(δ) terms make the use of that inequality more involved and we need to assumeδ sufficiently small (independent of ε). Due to the diffuse boundary conditions theanalysis of the remainder is here done for the scaling δ2 = γε for a suitably small γindependent of ε.

The main result of this paper is summarized in the following

Theorem 1.1. Assume δ2 = γε. Then, for ε and γ small enough, there exists apositive isolated L2-solution to the problem

vy∂F

∂y+

ε2

δ2C2σ(y)vx

(vx∂F

∂vy− vy

∂F

∂vx

)=

1εQ(F, F ), (1.23)

with diffuse reflection boundary conditions

F (∓π, v) = ∓M∓∫vy<0

dvvyF (∓π, v), ±vy > 0. (1.24)

Moreover, for q = 2 and q =∞,

‖ [F −M(1 + δvxU)]M−1 ‖q,2≤ cε, (1.25)

where U is the unique solution to (2.32) and ‖ · ‖q,2 is defined in (4.19).

Remark 1.1. The proof of the theorem shows that the rest term is of order ε3 inL∞.

2. Expansions. In this section we show how to compute the contributions Fn forn = 1, . . . , N in (1.19). A modified Hilbert expansion is used to compute the bulkterms. Since they violate the boundary conditions, we introduce boundary layercorrections essentially supported in thin layers (of size of the order of ε) near theinner and outer cylinders. Therefore, Fn is written as follows:

Fn = Bn + b+n + b−n , (2.1)

where Bn is a smooth function of y, while b±n are smooth exponentially fast decayingfunctions of the rescaled variables Y ± = ε−1(π∓ y), so that they are exponentiallysmall away from ±π.


2.1. The bulk expansion. In order to compute the expression of the Bn, wesubstitute (1.18) in (1.23). We ignore the terms b±n , because they are assumedexponentially small, and equate terms with the same power of ε up to the order N .We use the short notation

Lδf = 2Q(Mδ, f); (2.2)

N(f) = vx

(vx∂f

∂vy− vy

∂f

∂vx

). (2.3)

Moreover, since in this subsection the parameter δ is kept fixed, we omit the indexδ when there is no ambiguity. We get the following conditions:

LB1 = vy∂yM, (2.4)LB2 = vy∂yB1 −Q(B1, B1), (2.5)

LB3 = vy∂yB2 +1

C2δ2N(M)− 2Q(B1, B2), (2.6)

LBn = vy∂yBn−1 +1

C2δ2

n−3∑h=0

σ(h)N(Bn−3−h)

−∑

h,k≥1,h+k=n

Q(Bh, Bk), n = 4, . . . , N. (2.7)

In (2.7) B0 ≡ M and σ(h) are the coefficients of the ε-power series expansion ofσ(y):

σ(y) =∞∑h=0

εhσ(h)(y).

The appropriate functional space to solve the above equations is the the Hilbertspace H of the real measurable functions on the velocity space R3, equipped withthe inner product

(f, g) =∫

R3dvM−1(v)f(v)g(v). (2.8)

The operator L is defined in a suitable dense submanifold DL of H and satisfiesthe following properties:L1) L is symmetric and non positive: (f,Lg) = (g,Lf); (f,Lf) ≤ 0.L2) L has a 5-dimensional null space spanned by the collision invariants:

Null L = spanψ0, . . . , ψ4, (2.9)

with ψβ = χβM , β = 0, . . . , 4 and

χ0 = 1; χ1 = vx; χ2 = vy; χ3 = vz; χ4 =|v|2

2. (2.10)

The orthogonal projector on Null L is denoted P , while P⊥ = 1− P denotesthe projector on the orthogonal complement of Null L in H.

L3) The range of L is orthogonal to Null L: (ψα,Lf) = 0 for any α = 0, . . . , 4 andfor any f ∈ DL.

L4) The following decomposition holds:

Lf = −νf + Kf (2.11)

where K is a compact operator and ν a smooth function such that

ν0(1 + |v|) ≤ ν(v) ≤ ν1(1 + |v|) for all v ∈ R3. (2.12)


L5) If g ∈ P⊥H then, by the Fredholm alternative theorem and L4), there is aunique solution in P⊥H to the equation

Lf = g, (2.13)

which, with a slight abuse of notation, we denote by L−1g:

f = L−1g. (2.14)

Any solution in H of (2.13) can be written as

f = L−1g + f (2.15)

with f ∈ Null L.L6) Spectral inequality: there is a constant c > 0 such that

(f,Lf) ≤ −c(P⊥f, νP⊥f). (2.16)

By L3) in order to solve (2.4) we need to impose the compatibility conditionP (v · ∇M) = 0. It is immediate to check that this is true if and only if

∂yP = 0, (2.17)

whereP = ρT, ρ = 1 + δr, T = 1 + δτ. (2.18)

This is just the second of (2.31).If this condition is satisfied, then

vy∂yM = δ(A∂yU + B∂yτ), (2.19)

where

B = vxvyM,

(2.20)

A =v2 − 5T

2T 2vyM

are in P⊥H. We define A and B as the solutions in P⊥H of the equations

LA = A; LB = B. (2.21)

Therefore, by L5)A = L−1A, B = L−1B. (2.22)

Henceforth,

B1 = δ(B∂yU + A∂yτ) +M

(ρ1

ρ+v · uT

+v2 − 3T

2T 2τ1

):= B⊥1 +B

‖1 , (2.23)

with ρ1, τ1 and u to be determined by the other conditions.In order to solve (2.5), we need to impose the orthogonality of the right hand

side of (2.6) to the null space of L.A standard computation shows that this is equivalent to the conditions

∂y(ρuy) = 0ρ (uy∂yU) = ∂y(η∂yU), (2.24)∂y(Tρ1 + ρτ1) = 0,52ρuy∂yτ = ∂y(κ∂yτ) + δη(∂yU)2,


with

η = −∫

R3dvBB, κ = −

∫R3dvAA.

Then we can compute the part B⊥2 of B2 as before:

B2 = −L−1Q(B1, B1) + L−1P⊥(v · ∇B1) +B‖2 , (2.25)

with

B‖2 = M

(ρ2

ρ+v · u2

T+v2 − 3T

2T 2τ2

), (2.26)

and ρ2, u2 and τ2 to be determined.The same procedure is applied to solve the equation for B3, (2.6). The compat-

ibility condition is

P(vy∂yB2 +

1C2δ2

N(M))

= 0. (2.27)

This implies several conditions. We write explicitly only the following:

ρuy∂yuy + ∂yP2 −1C2

ρU2 = ∂y

(η∂yuy +

δ2

P

[σ1(∂yτ)2 + σ2(∂yU)2

]). (2.28)

Here σi are some suitable transport coefficients of higher order whose explicit ex-pression is given for example in [17].

Note that the term in U2 in equation (2.28) derives from the contribution∫dvvyN(M) = O(δ2). The result in (2.28) is independent of δ because of the scaling

(1.15) and hence it persists in the limit δ → 0.The equations written so far are just the system (2.31). They represent a system

in the unknown functions U , uy, τ and r, which does not include any of the extrafuntions ρ1, τ1 etc, which also have to satisfy some extra conditions, for examplethe Boussinesq condition to the first order in ε.

We do not give the explicit conditions which follow in a rather standard way. Itis clear that the above procedure can be continued to any specific order. In thispaper it will be truncated at N = 5. Note that at each step the solution is givenup to the choice of five arbitrary functions which are fixed in the subsequent steps.In particular the last term of the truncated expansion, BN , is determined up to itshydrodynamic part which is arbitrary. We will take advantage of this arbitrarinesswhen dealing with the equation for the remainder.

2.2. The boundary layer expansion. We need to include in our scheme a bound-ary layer expansion, because the bulk terms B1 do not satisfy the diffuse reflectionboundary condition. For example, it is immediate to check from equation (2.23)that B1 cannot be proportional to the Maxwellian M± at the boundaries. Thereforewe introduce the corrective terms b±1 , with a fast dependence on y, so that they aresensibly different from 0 only close to the boundary. To achieve this, b±1 is assumedto be a smooth function of the variable Y ± = ε−1(π ∓ y). We define b±1 as thesolution to the following equation:

vy∂b±1∂Y ±

+ε3

C2δ2σ(∓(εY ± − π))N(b±1 ) = L±b±1 + L±ϑ b

±1 , (2.29)

with L±g = 2Q(M±, g), M± = M(1, 1, (δU±, 0, 0); v). Here σ = σϕ, with ϕ asmooth cutoff function

ϕ(y) =

1 y ∈ [0, ζ],0 y > 2ζ


for some ζ > 0 and with uniformly bounded derivatives. The operator L±ϑ is definedin the same way as L±, but we replace the hard spheres collision cross sectionB(n, V ) with ϑB(n, V )2. The reason for introducing this unphysical operator is dueto a technical difficulty which will be discussed in the next section. The parameterϑ is chosen as ϑ = ε3

C2δ2 , and the contribution from L±ϑ , which should not be there,will be subtracted in the next order of the boundary layer expansion. We prescribevanishing mass flux at the boundary:

m±1 =∫

R3dvvy b

±1 (0, v) = 0.

This equation has to be solved with prescribed incoming data at Y ± = 0:

b±1 (0, v) = h±1 (v), for vy > 0.

The incoming boundary data are chosen in such a way to compensate the fact thatB⊥1 is not proportional to a Maxwellian at the boundary: h±(v) = −B⊥1 (±π). Thesolution to the Milne problem in general has a finite, but not vanishing limit atinfinity, achieved exponentially fast. Let it be denoted by b±1,∞. It belongs to thenull space of L±. The non vanishing of b±1,∞ is not good to our purposes becausethis contributes to the solution in the bulk. Therefore we define b±1 = b±1 − b

±1,∞. In

this way we ensure the decay at infinity, but b±1 do not satisfy any more the equation(2.29) , because a term of the form ε3

C2δ2 σ(∓(εY ±−π))N(b±1,∞) appears in the righthand side. We will compensate it with a term in the next order of the boundarylayer expansion. We are not yet done, because the boundary value of F±1 would stillbe incorrect for the term −b±1,∞ which is not Maxwellian. However, we have not

yet fixed the boundary values of B‖1 and we can use them to compensate it, sinceit is in the null space of L±. Finally, note that, on each boundary, the boundaryvalue correction due to the other boundary is not zero, but exponentially small inε−1. This will be compensated in the remainder. In conclusion f1 = B1 + b+1 + b−1satisfies the diffuse reflection boundary conditions up to terms Ψ±1 exponentiallysmall in ε−1.

The equation (2.29) is a special case of the Milne problem we discuss in the nextsection. We note however that in the standard Milne problem the second term inthe left hand side of (2.29) is absent. When in the equation there is a force term, asin the present case and in [13, 1, 2], although very small, the lack of regularity in thevelocity of the solution to the Milne problem for vy = 0 (the derivative ∂vy

b±1 doesnot exist for vy = 0 at the boundary), does not allow us to include them in higherorder terms of the expansion. Indeed we need to keep it in (2.29) which will besolved in a suitably weak sense, because we cannot afford to have any vy derivativeof b±1 present in the expansion. This problem was already present in the case of theBenard problem where the force derives from a potential and the solution is givenin [10].

The corrections to Bn, for n > 1 will be b±n solving a similar equation:

vy∂b±n∂Y ±

+ε3

C2δ2σ(∓(εY ± − π))N(b±n ) = L±b±n + L±ϑ b

±n + S±n , (2.30)

with prescribed incoming data at Y ± = 0:

b±n (0, v) = h±n (v), for vy > 0,


vanishing mass flux at the boundary:

m±n =∫

R3dvvy b

±n (0, v) = 0

and then define b±n = b±n − b±n,∞. The incoming boundary data are chosen in sucha way that fn = Bn + b+n + b−n satisfies the diffuse reflection boundary conditionsup to terms exponentially small in ε−1, Ψ±n . The source term S±n has the followingform, for n > 1:

S±n = −L±ϑ b±n−1 +

∑h,k≥1,h+k=n

[2Q(Bh, b±k ) +Q(b±h , b

±k ) +Q(b±h , b

∓k )

− ε2

C2δ2σc(∓(εY ± − π))N(b±n−1)− ε2

C2δ2σ(∓(εY ± − π))N(b±n−1,∞)

]with σc = σ(1−ϕ) and b±0 = 0 and the property

∫dvS±n (y, v) = 0. The existence of

the solutions to (2.29), (2.30) with the prescribed conditions follows from Theorem3.1, given in next section, via a procedure which is the same used in [12, 13, 1, 2].We do not repeat it here and refer to those papers for details.

2.3. Hydrodynamical expansion. In this section we compare the solution of thestationary 1-d equations for δ > 0 with the solution of the limiting equations forδ = 0. The former are

∂y(ρuy) = 0,∂y[(1 + δρ)(1 + δτ)] = 0,ρuy∂yU = ∂y(η∂yU),52ρuy∂yτ = ∂y(κ∂yτ) + δ(η∂yU)2, (2.31)

ρuy∂yuy + ∂yP2 −1C2

ρU2 = η0∂2yuy

+∂y

(ηδ∂yuy +

δ2

P

[σ1(∂yτ)2 + σ2(∂yU)2

]),

with ρ = 1 + δr and the boundary conditions

uy(±π) = 0; Ux(±π) = U±, τ(±π) = r(±π) = 0.

In the limit of vanishing δ the velocity uy and τ are identically zero and

η0∂2yU = 0, (2.32)

∂yP2 −1C2

ρU2 = 0

whose solution is the laminar field U = U−+ β(y+ π), with β = (2π)−1(U+−U−).The first equation in (2.31) implies, by using the boundary conditions for uy,

ρuy = 0. Since ρ = 1+δr for δ small is strictly larger than zero, it has to be uy = 0.The equations reduce to

∂y(η∂yU) = 0,

∂y(κ∂yτ) + δ(η∂yU)2 = 0, (2.33)

∂yP2 −1C2

ρU2 =δ

P∂y(σ1(∂yτ)2).

The first two equations decouple from the third and can be solved to find U and τ .Then the last one gives P2.


We define U = U − U . We have

∂y(η∂yU) + β∂yη = 0,

∂y(κ∂yτ) + δη(β + ∂yU)2 = 0, (2.34)

U(±π) = 0; τ(±π) = 0.

The functions η and κ are smooth functions of the temperature. The solutionsare constructed by an iterative procedure. We therefore assume that ‖τ‖∞ < 1 .In consequence ‖η‖∞ and ‖κ‖∞ are uniformly bounded for δ < 1. Moreover, for δsufficiently small, ‖η‖∞ > η0

2 and ‖κ‖∞ > κ02 with η0 and κ0 the values of η and κ

for τ = 0. Finally, note that ∂yη = δη′∂yτ with η′ = ∂η∂T uniformly bounded. By

multiplying the first of (2.34) by U and the second by τ and integrating in y, afteran integration by parts we get:

η0

2‖∂yU‖2 ≤ cδ‖U‖‖∂yτ‖ ≤ cδ‖∂U‖‖∂yτ‖,

κ0

2‖∂yτ‖2 ≤ cδ‖τ‖∞(β2 + ‖∂U‖2) ≤ cδ‖∂τ‖(β2 + ‖∂U‖2).

Therefore

‖∂yU‖ ≤ cδ‖∂yτ‖,‖∂yτ‖ ≤ cδ(β2 + ‖∂yU‖2).

The above inequalities imply that ‖∂yU‖ and ‖∂yτ‖ are O(δ). In particular‖∂yτ‖∞ < 1 for δ sufficiently small. We omit the proof of the convergence of theapproximating sequence, which follows along the same lines.

We conclude with the following

Theorem 2.1. If δ is sufficiently small, the equations (2.31) have a uniqueC∞(−π, π) stationary solution which differs from the laminar solution (namelyU = U(y), τ = 0) for O(δ).

Call (ρδ, Uδ, uδy, τδ) the solution of (2.31) (remember uδy = 0) and (1, U, 0, 1) the

solution of the equations for δ = 0 with U the unique solution of (2.32).Let Mδ = M(ρδ, (δUδ, 0), τ δ) and M0

δ = M(1, (δU, 0), 1). Then

Corollary 2.2. For q = 2,∞

‖M−1[Mδ −M0δ ] ‖q,2≤ cδ2, ‖M−1[M0

δ −M ] ‖q,2≤ cδ.

2.4. Estimates for the expansion. The properties of Fn’s constructed in thissection are summarized in the following theorem:

Theorem 2.3. The functions Fn, n = 1, . . . , 5 and Ψn,ε can be determined so asto satisfy the boundary conditions

Fn(∓π, v) = M∓(v)∫wy≶0

|wy|[Fn(∓π,w)−Ψn,ε(∓π,w)]dw

+Ψn,ε(∓π, v), vz ≷ 0,

and the normalization condition∫

R3×[−π,π]dvdyFn = 0, so that the asymptotic ex-

pansion in ε for the stationary problem (1.23), truncated to the order 5 is given


by

Φ =5∑

n=1

εnFn(y, v).

The functions Fn’s satisfy the conditions

‖M−1Fn ‖2,2<∞, ‖M−1Fn ‖∞,2<∞ , n = 1, . . . , 5.

The functions Ψn,ε are such that ‖Ψn,ε‖q,2,∼, q = 2,∞, are exponentially small asε→ 0 and

∫R3 dvvyΨn,ε = 0.

Proof. It follows as in [1, 2], using the Theorem 3.1 in next section.

3. Milne problem. In this section we deal with the Milne problem

vy∂g

∂Y+Gω(Y )Ng = Lg + S,

g(0, v) = h(v), vy > 0, (3.1)∫dvvyM(v)g(0, v) = 0,∫dvS(y, v)M = 0

for S and h prescribed. Here M is the Maxwellian with T = 1, ρ = 1 and u =(U, 0, 0);

Lf = 2M−1[L(Mf) + Lϑ(Mf)], Nf = M−1N(Mf).and ω(Y ) a compactly supported smooth function. Moreover, we assume that thereis c > 0 such that ∫

vy>0

dvvyM(v)h2(v) < c. (3.2)

Note the explicit expression of Nf :

Nf = v2x

∂f

∂vy− vxvy

∂f

∂vx− Uvxvyf. (3.3)

The results will be applied with G = ε3

δ2C2 , ϑ > GU, ω = σ and U = δU±. Theprocedure is the same used in [10]: we construct the solution in a slab of size ` withreflecting boundary condition g(`, Rv) = g(`, v), Rv = (vx,−vy, vz) and obtainestimates uniform in `, then we take the limit ` → ∞. As in [10], the main pointis to discuss the case S = 0. The assumptions on S will be given in Theorem 3.1below. We only point out the differences with [10]. We write g = q + w withq ∈ Null L and (q, w) = 0. We set χ0 = 1, χ1 = vx − U, χ2 = vy, χ3 = vz,χ4 = 1

2 [(vx − U)2 + v2y + v2

z ], and q =∑4α=0 bα(Y )χα. Moreover, ( · , · ) denotes the

inner product on L2(R3,Mdv).Note that b2 = 0. Indeed, by multiplying (3.1) by M and integrating in dv, we

get∂

∂Y(vy, g) +Gω(vy, g) = 0,

because(1, Ng) =

∫dvgMvy.

Therefore, with Ω(Y ) =∫ Y

0dY ′ω(Y ′), we have

b2(Y ) = (vy, g)(Y ) = exp−G[Ω(`)− Ω(Y )](vy, g)(`) = 0


because (vy, g)(`) vanishes by the reflecting boundary conditions at Y = `.As a consequence, q =

∑α 6=2 bα(Y )χα. Moreover, (vyχα, χβ) = 0 for α, β 6= 2.

Therefore (vyq, q) = 0.Set Iα = (vyχα, g) = (vyχα, w) for α 6= 2. The functions Iα satisfy the following

equations:∂

∂YIα = ω

∑β 6=2

Sα,βIβ ,

with

Sα,β = δα,β + (δα,1 − Uδα,4)δ1,β ,

for α, β 6= 2. Indeed,

(χα, Nχβ) =∫dvMχα

(Uvxvy(δβ,4 − 1)− vxvyδβ,1

),

which is odd in vy for α 6= 2. On the other hand,

(χα, Nw) =∫dv[vyMχαw +Mvyvxw(δα,1 − Uδα,4)

].

Therefore, with I = (Iα)α6=2, we have

I(Y ) = I(`) exp(Ω(`)− Ω(Y ))S,

which implies

I(Y ) = 0

by the reflection boundary condition at Y = `.We take the inner product of the first equation in (3.1) with g. By using (3.3),

(g,N(g)) =12

(vyg, g)− 12U(vxvyg, g),

and so we obtain:

12∂

∂Y(vyg, g) +

12Gω(vyg, g) = (w,Lw) +

12GUω(vxvyg, g)

= (w,Lw) +12GUω(vxvyw,w) +GUω(vxvyw, q). (3.4)

because (vxvyq, q) = 0. The last but one term in the above Green identity is handledby adding in the cross section in the linearized Boltzmann operator the unphysicalterm ϑB( · , · )2. Indeed, in this case the inequality (2.16) holds with ν replaced byν + ϑν, with ν satisfying the inequalities (2.12) and ν such that

ν0(1 + |v|)2 ≤ ν(v) ≤ ν1(1 + |v|)2, v ∈ R3 (3.5)

for suitable constants ν0 and ν1. This allows to control the term GUω2 (vxvyw,w) as

long as we have ϑ > GU2 .

So we have only to worry about the term ωGU(vxvyq, w) for which we use the bound

ωGU|(vxvyq, w)| ≤ 12ωGU‖w‖2 +

c

2ωGU‖q‖2.


We set A = (vyg, g). Note that A(0) < c by (3.2). We need upper and lowerbounds on A(0). We can write

A(0) = A(`) expG2

(Ω(`)+∫ `

0

dY expG2

(Ω(`)− Ω(Y )[− (w, (L+

12UGωvxvy)w)(Y )−GUω(Y )(vxvyq, w)

](3.6)

By the reflecting boundary conditions, A(`) = 0. Moreover, (w, (L+ 12GUωvxvy)w) ≥

−(w, νw) for ϑ > 12GU. We now need to estimate the last term. We use the follow-

ing bound proved later:

‖q(Y )‖ ≤√c+ |A(0)|+ c‖

√νw‖(Y ) +

∫ Y

0

dY ′‖√νw‖(Y ′). (3.7)

We use the bound∫ `

0

ωdY

[∫ Y

0

||w||(Y ′)dY ′]2

≤∫ `

0

ω(Y )Y dY∫ `

0

||w||2(Y ′)dY ′

≤ c∫ Y

0

||w||2(Y ′)dY ′. (3.8)

Then we plug (3.7) in (3.6) and use the spectral inequality, to get the followingbound for |A(0)|, using the fact that ω is compactly supported,

|A(0)| ≤ G(c+ c|A(0)|),which implies the boud on |A(0)| for G sufficiently small. Using this one can con-clude that ∫ `

0

dY ‖w(Y )‖2 < c (3.9)

uniformly in `.We need to prove the bound (3.7). Let βα = (v2

yχαq) for α 6= 2. Since βα =∑γ 6=2Aα,γbγ , with Aα,γ = (v2

yχα, χγ) a positive non singular matrix, to estimateβα is equivalent to estimate ‖q‖. The equation for βα is obtained by taking theinner product of the first equation in (3.1) with vyχα. The result is

∂βα∂Y

= Gω∑γ 6=2

Bα,γβγ + Dα + (vyχα, Lw),

with

Dα = − ∂

∂Y(v2yχα, w)−Gω(vyχα, Nw)

and

Bα,γ = Gω[δα,γ(1 + U(δα,1 −Uδα,4)− (v2

xχα, χγ) + (δα,1 − Uδα,4)(v2x, v

2y)A−1

α,γδ1,γ

].

An integration by parts shows that |(vyχα, Nw)| ≤ c‖w‖. The rest of the argumentis as in [10]. The only difference is in the estimate of βα(0). We have

|(v2yχα, g)(0)| ≤ (|vy|g, g)1/2(|vy|3, |χα|).

(|vy|g, g)(0) =∫vy>0

vyh2 −

∫vy<0

vyg2 = 2

∫vy>0

vyh2 −A(0).

By using (3.2) we then get (3.7).


To get estimates uniform in ` also for q we take the scalar product of the firstequation in (3.1) equation by L−1(χαvy). The term on the right hand side is(L−1(χαvy), Lw), which is zero by the orthogonality property. We get then anequation for Θα = (vyL−1(χαvy), q) whose solution is

Θ(Y ) = e−R Y0 dsGω(s)(GQ−1)(L−1(vyχ), vyg)(0)− (L−1(vyχ), vyw)(Y )

+∫ Y

0

dte−R Y

tdsGω(s)(GQ−1)D(t)

where G and Q are suitable matrix, with Θα = Qαγbγ , Q invertible, and

Dα(Y ) = −Gω(Y )[(L−1(vyχα), Nw)− ∂

∂Y(L−1(vyχα), vyw)

].

By Schwartz inequality, boundedness of L−1 and the fact that ω has compactsupport, the last integral is finite. Moreover, ‖ w ‖ vanishes at infinity and the firstterm on the right hand side is finite. This implies that there exists a finite limit atinfinity, Θ∞, of Θ and there is Y0 such that for Y > Y0 we have

|Θ(Y )−Θ∞|2 ≤ c||w||2(Y ).

By the argument in [10] we get also for Y > Y0

|b(Y )− b∞|2 ≤ c||w||2(Y ).

The other arguments in [10] can be adapted in a similar way. Using above esti-mates, the exponential decay of ‖w‖ and |bα − b∞α | is established. The propertiesof the derivatives are also obtained with the method presented there, with a minormodification due to the special structure of the force in this case. Indeed, the vxand vy derivatives of g satisfy in this case a coupled system of equations. Therefore,they are both controlled only away from the boundary.

To state the final theorem, we define the norms

‖ f‖q,2,θ =

(∫R3dvM(v)

(∫ ∞θ

dY |f(Y, v)|q) 2

q

) 12

, (3.10)

for θ ≥ 0.

Theorem 3.1.1) Suppose that for some β > 0

‖ eβY S ‖2,2,0<∞, ‖ eβY S ‖∞,2,0<∞ .

Then there is a unique solution g ∈ L2(R3, L∞(R+)) ∩ L2(R+ × R3) to the Milneproblem (3.1). Moreover there exist constants c and c′ such that g verifies theconditions: ∫

R3Mgdv = 0, g∞ ∈ Null L,

‖ eβ′Y (g(Y, v)− g∞(v)) ‖2,2< c, ‖ eβ

′Y (g(Y, v)− g∞(v)) ‖∞,2< c

for any β′ < c′.2) Suppose that for ` ≥ 1, θ > 0 and i = 1, . . . , 3∫

vy>0

M∣∣∣∂`h∂vì

∣∣∣ <∞, ∥∥∥eβY ∂`S∂vì

∥∥∥2,2,θ

<∞,∥∥∥eβY ∂`S

∂vì

∥∥∥∞,2,θ

<∞.


Then there are constants c′ and c` such that∥∥∥eβ′Y ∂`g∂v`i

∥∥∥2,2,θ

< c`,∥∥∥eβ′Y ∂`g

∂v`i

∥∥∥∞,2,θ

< c`.

for any β′ < c′.

4. The remainder.

4.1. Equation for the remainder. It is immediate to check that the remainderR in (1.18) has to satisfy the following equation:

vy∂R

∂y+

ε2

δ2C2σ(y)vx

(vx∂R

∂vy− vy

∂R

∂vx

)=

1ε

[LδR+2Q(Φ,R)

]+εQ(R,R)+A,

(4.1)

with the inhomogeneous term A given by

A =∑

h,k≥1,h+k>N

εh+k−2Q(Fh, Fk)− εN−1vy∂yBN−

− ε2

C2δ2

∞∑h=N−2

N∑n=0

εh+nσ(h)N(Bn)− ε2

C2δ2

N−3∑h=0

εhσ(h)N∑

n=N−2

εnN(Bn−h)

− εN∑±

[L±ϑ b

±N +

ε3

C2δ2σc(∓(εY ± − π))N(b±N,∞)

], ϑ =

ε3

C2δ2(4.2)

We remind that Mδ = M(1 + δr, 1 + δτ, (δU, 0, 0); v), where (r, τ, (U, 0, 0)) is thesolution of (2.31). We keep now the dependence on δ and denote by M the standardMaxwellian. We write R = MR and denote by L the operator

Lf = M−1LMf. (4.3)

We use the Hilbert space H of the measurable functions on the velocity space R3

with inner product

(f, g) =∫

R3dvM(v)f(v)g(v). (4.4)

Note that this involves M , while the one considered in Section 2, (2.8) involvedM−1δ . The operator L has the same properties already mentioned for Lδ and we do

not repeat them here. We just note that the null space of L is given by the spanof the χβ ’s defined in (2.10). We still denote the projector on it by P and on itsorthogonal complement by P⊥ = 1− P .

Moreover, we write

M−1Lδf = Lf +M−1[Lδ − L]Mf. (4.5)

We also use the notation

J(f, g) = M−1Q(Mf,Mg). (4.6)

The following operator will play a major role:

LJf = Lf + Γf, Γf = 2J(W,Pf), W =Mδ −M

M+

ΦM. (4.7)

By Theorem 2.3 and Corollary 2.2, W = O(δ + ε) and hence the operator Γ isO(δ + ε).

In [2] it is proved the following


Theorem 4.1. The operator LJ has the following properties:1. The null space of LJ is the span of the functions

ψJβ = χβ − L−1Γχβ , β = 0, . . . , 4; (4.8)

PJ denotes the projection on it.2. The null space of its adjoint L∗J is the same as L.3. If δ and ε are sufficiently small, the following spectral inequality holds:

(f, LJf) ≤ −c((1− PJ)f, ν(1− PJ)f) (4.9)

for a suitable positive constant c.

Using these notation, we write the equation (4.1) for the remainder as follows:

vy∂R

∂y+ε2

c2σ(y)vx

(vx∂R

∂vy− vy

∂R

∂vx

)=

1εLJR+

1εH1R+ εJ(R,R) + a, (4.10)

with a = M−1A andH1f = 2J(W, (1− P )f). (4.11)

Note that, thanks to the arbitrariness left in the determination of the FN , we canassume that

Pa = 0. (4.12)

The equation for the remainder has to be solved with the boundary conditions:

R(−π, v) = α−(R)M−1M− −1ε

Ψ(−π, v), vy > 0

(4.13)

R(π, v) = α+(R)M−1M+ −1ε

Ψ(π, v), vy < 0,

where

MΨ(∓π, z, v, t) =n∑n=1

εnΨn,ε(∓π, z, v, t) (4.14)

which are exponentially small in ε−1 because of Theorem 2.3, and

α−(R) = −∫vy<0

dvvyM [R(−π, v) +1ε

Ψ(−π, v)],

(4.15)

α+(R) =∫vy>0

dvvyM [R(π, v) +1ε

Ψ(π, v)].

4.2. Estimates for the remainder. We will proceed with the construction of thesolution by iteration, based on estimates for a linearized problem where the nonlinear term J(R,R) is computed at the previous step of the iteration. The genericterm of the iteration will then satisfy a linear equation of the type

vy∂R

∂y+

ε2

δ2C2σ(y)vx(vx

∂R

∂vy− vy

∂R

∂vx) =

1ε

[LJR+H1(R)] + g. (4.16)

At the step n of the iteration the term g will be replaced by a + εJ(Rn−1, Rn−1),therefore we assume

Pg = 0. (4.17)


The boundary conditions are

R(∓π, z, v) =M∓M

∫∓wy>0

(R(∓π,w) +1ε

Ψ(∓π,w))|wy|Mdw

− 1ε

Ψ(∓π,w)), ±vy > 0. (4.18)

In order to simplify the argument we also assume that the velocity of the innercylinder vanishes, so that M− is the standard Maxwellian, up to the normalization.The argument can be easily adapted to the general case.

We first consider the linear problem with g given and will get L2 bounds. Thisis done in two steps: first, we get a control of the non-hydrodynamic part in termsof the hydrodynamic one. Second, we get an estimate of the hydrodynamic part interms of the non-hydrodynamic one and then a bound for R in terms of the knownterm g.

The norms used below are defined as follows:

‖ f‖q,2 =

∫R3dvM(v)

(∫[−π,π]

dy|f(y, v)|q) 2

q

12

. (4.19)

Defining the ingoing velocity spaces R3± at y = ∓π as the sets v = (vx, vy, vz) such

that vz ≷ 0

‖ f ‖q,2,∼= sup±

(∫R3±

dv|vy|M(v) (|f(∓π, v)|q)2q

) 12

. (4.20)

We use the traces

γ±f =

f |y=−π, if vy ∈ R3

±,

f |y=π, if vy ∈ R3∓.

(4.21)

Note that the norm ‖ · ‖2,2,∼ is defined only for incoming velocities. In the sequel,with an abuse of notation we will denote by ‖ γ−f ‖2,2,∼ the ‖ · ‖2,2,∼-norm ofSγ−f , where S is the reflection of the y component of the velocity.

We now establish the lemmas which allow to bound the solution to (4.16) inthese norms.• Step 1

Lemma 4.2. Assume δ2 = εγ. Then, for γ small enough and for any η > 0, thesolution of (4.16) satisfies

1ε‖ ν 1

2 (I − PJ)R ‖22,2 ≤ c[(εγ + εη) ‖ PJR ‖22,2 +

1ηε‖ PJg ‖22,2 (4.22)

+1ηε3‖ Ψ ‖22,2,∼ +ε ‖ (I − PJ)g ‖2,2

]2,

‖ γ−R ‖22,2∼ ≤ c[η ‖ PJR ‖22,2 +

1ηε2‖ (I − PJ)R ‖22,2 +ε2 ‖ (I − PJ)g ‖22,2

+ ‖ PJg ‖22,2 +1ηε3‖ Ψ ‖22,2,∼

]. (4.23)


Proof. Define

k(y) = exp(∫ y

−π

ε2

δ2C2σ(q)dq

). (4.24)

Multiply (4.16) by 2RMk(y) and integrate over v. We get

∂

∂y(vyR, kR) = 2k[(R,LJR) + (R,H1(R)) + (R, g)].

Introduce R = R√k(y),g = g

√k(y) and Ψ = Ψ

√k(y).

∂

∂y(vyR, R) = 2

[(R, LJ R) + (R,H1(R)) + (R, g)

].

We bound all the terms in the r.h.s. in the following way: We write

LJ R =12LR+

12LR+ ΓR =

12LR+ LJ R.

We notice that LJ also satisfies the same spectral inequality as LJ but with adifferent constant and for smaller δ and ε. We use this spectral inequality and theone for L to get

(R, LJ R) ≤ −c ‖ ν 12 (I − P )R ‖22 −c ‖ ν

12 (I − PJ)R ‖22 .

We use the fact that H1 depends only on (I − P )R, J(·, ·) is orthogonal to the nullspace of L and that H1(R) is of order δ + ε to get the bound

(R,H1(R)) ≤ c(δ + ε) ‖ ν 12 (I − P )R ‖22,

so that(R, LJ R) + (R,H1(R)) ≤ −c ‖ ν 1

2 (I − PJ)R ‖22 .Moreover,

(R, g) = ((I − PJ)R, (I − PJ)g) + (PJ R, PJ g) ≤ c[ 1εη‖ (I − PJ)R ‖22

+εη ‖ (I − PJ)g ‖22 +ηε ‖ PJ R ‖22 +1εη‖ PJ g ‖22

],

with η > 0 an arbitrary number.Next, integrate over [−π, π]× R3. Putting all the estimates together,

−B + c11ε‖ ν 1

2 (I − PJ)R ‖22,2≤ c[ε ‖ (I − PJ)g ‖22,2

+1εη‖ PJ g ‖22,2 +εη ‖ PJ R ‖22,2

], (4.25)

where B := (vy, R2(−π, v))− (vy, R2(π, v)).Following the argument in [13] one can show that the first term in B is nonpositive

and the second can be bounded as

−∫

R3MvyR

2(π, v)dv ≤ c(δ2 + εη)∫vy>0

M |vy|R2(π, v)dv

+1ε3η

∫vy>0

M |vy|Ψ2(π, v)dv. (4.26)

Replacing in (4.25) we get


1ε‖ ν 1

2 (I − PJ)R ‖22,2≤ c[ε ‖ (I − PJ)g ‖22,2 +

1εη‖ PJ g ‖22,2 (4.27)

+ηε ‖ PJ R ‖22,2 +(δ2 + εη)∫vy>0

M |vy|R2(π, v)dv]

+1ε3η

∫vy>0

M |vy|Ψ2(π, v)dv.

To estimate the outgoing integral of R in the right hand side, multiply (4.16) byMR√k, integrate in velocity over the region vy ≥ q, then over space using a smooth

cut-off function χ(y) which is zero close to y = −π, and equal to one close to π,and finally over q for q0 ≤ q ≤ 0 for |q0| small enough. Since we do not integrateover all v ∈ R3 we cannot use the spectral inequality to control the terms involvingLJ and H1, but we use only the boundedness of LJ . We get∫ 0

q0

dq

∫vy≥q

dvMvyR2(π, v) ≤ c|q0|

[(η + δ2 + ε2) ‖ PJ R ‖22,2

+1ηε2‖ (I − PJ)R ‖22,2 +ε2 ‖ (I − PJ)g ‖22,2 +

1η‖ PJ g ‖22,2

](4.28)

+∫ 0

q0

dq

∫vy≥q

dvdyMχ′(y)vyR2(y, v).

The term on the l.h.s. equals

|q0| ‖ γ−R(π) ‖22,2∼ +∫ 0

q0

dq

∫q≤vy<0

dvvyMR2(π, v),

whereγ−R(π) = R(π, v), vy > 0.

We move the second term in the previous expression to the r.h.s. of (4.28) andbound it, by replacing R by the ingoing boundary data, as∣∣∣∣∣∫ 0

q0

dq

∫0≤vy<q

1Mvydv

[M+

∫wy≥0

dwwyM(R(π,w) +

1ε

Ψ(π,w))

−M 1ε

Ψ(π, v)]2∣∣∣∣∣ ≤ c(q0)

[‖ γ−R(π) ‖22,2∼ +

1ε2‖ Ψ ‖22,2,∼

], (4.29)

with c(q0) = o(|q0|). We replace this estimate in (4.28) and divide both sides by|q0|

‖ γ−R(π) ‖22,2∼≤ c[(η + δ2 + ε2) ‖ PJ R ‖22,2 +

1ηε2‖ (I − PJ)R ‖22,2

+ε2 ‖ (I − PJ)g ‖22,2 +1η‖ PJ g ‖22,2 +

1ε2‖ Ψ ‖22,2,∼

]. (4.30)

We have estimated the term

−∫ q0

0

dq

∫vy≥q

dvdyMχ′(y)vyR2(y, v)

as η1|q0| ‖ R ‖22,2, with η1 small, by using a suitable cut-off function χ with positivederivative χ′ < 1 and the bound −vy ≤ −q in the integral.


Finally, we use the above bound for ‖ γ−R ‖22,2∼ in (4.27) to get,

c11ε‖ ν 1

2 (I − PJ)R ‖22,2 ≤ [εγ + εη] ‖ PJ R ‖22,2 +1εη‖ PJ g ‖22,2

+1ε3η‖ Ψ ‖22,2,∼ +ε ‖ (I − PJ)g ‖22,2 . (4.31)

To deal with the two terms

δ2η ‖ PJ R ‖22,2,δ2

ηε2‖ (I − PJ)R ‖22,2 .

we had to choose δ2 = εγ, with γ a small number. With this choice the secondterm gets a factor γ

ηε which can be absorbed on the l.h.s for γ small and the firstone gets a factor εγη. We remark that this last term comes from the estimate ofthe outgoing flux and is due to the fact that the Maxwellians on the two boundariesdiffer by a factor δ.

Now it is easy to bound also γ−R(−π) by similar arguments and get

‖ γ−R(π) ‖22,2∼ ≤ c[η ‖ PJ R ‖22,2 +1ηε2‖ (I − PJ)R ‖22,2 +ε2 ‖ (I − PJ)g ‖22,2

+1η‖ PJ g ‖22,2 +

1ε2‖ Ψ ‖22,2,∼

]. (4.32)

Finally, we notice that the norms involving R, g are equivalent to the ones forR, g since k(y) defined in (4.24) is a function uniformly bounded in ε.

• Step 2

We will provide an a priori estimate for the hydrodynamic part of R. Theequation for the remainder is

vy∂R

∂y+

ε2

δ2C2σ(y)vx(vx

∂R

∂vy− vy

∂R

∂vx) =

1ε

[LJR+H1(R)] + g (4.33)

with the boundary conditions

R(∓π, v) = −1ε

Ψ(∓π,w)+M∓M

∫∓wy>0

(R(∓π,w)+1ε

Ψ(∓π,w))|wy|Mdw, ±vy > 0

(4.34)where now in the definition of LJ the function Φ is determined by Theorem 2.3.

We simplify the problem by changing the boundary conditions in π from diffusiveto given ingoing data. The following lemma states the equivalence between the twoproblems.

Lemma 4.3. Equation (4.33) with the new boundary conditions

R(−π, z, v) =M−M

∫−wy>0

(R(−π,w) +1ε

Ψ(−π, z, w))|wy|Mdw,

−1ε

Ψ(−π,w), vy > 0 (4.35)

R(π, z, v) =M+

M

∫wy>0

1ε

Ψ(π,w)|wy|Mdw − 1ε

Ψ(π, v), vy < 0

has the same solution as problem (4.33)-(4.34)


Proof. We start by noticing that existence and uniqueness for the new problem areclassical. The main point is that the new boundary condition implies∫

vy>0

R(π, v)vyMdv = 0.

In fact, by multiplication of (4.33) by k(y)M and integration over v ∈ R3, since∫gMdv = 0, we get

∂

∂y

(k(y)

∫vyMRdv

)= 0, (4.36)

which implies

k(π)∫vyMR(π, v)dv = k(−π)

∫vyMR(−π, v)dv.

Hence, since∫

R3 vyMR(−π, v)dv = 0, we have also∫vyMR(π, v)dv = 0. We write

the integral in the left hand side by using the boundary condition (4.35)

0 =∫

R3vyMR(π, v)dv =

∫vy<0

vyM+dv

∫wy>0

1ε

Ψ(π,w))wyMdw (4.37)

+∫vy>0

vyMR(π, v)dv −∫vy<0

vy1ε

Ψ(π, v) =∫vy>0

vyMR(π, v)dv

because∫vy<0

vyM+dv = −1 and∫

R3 vyΨ(π, v) = 0.Hence,

∫vy>0

R(π, v)vyMdv = 0 and this implies that the unique solution R of thenew problem also satisfies the old boundary conditions (4.34).

Remark 4.1. Note that the condition∫

R3 R(π, v)vyMdv = 0 is crucial to makethis argument work. However, while the old boundary conditions are constructedin such a way that this is true, the new boundary conditions do not ensure thatit is automatically satisfied. It is the conservation law (4.36) which ensures thevanishing of the outgoing flux also with the new boundary conditions.

Lemma 4.4. If∫Mgdv = 0, then the solution of (4.33) satisfies

‖ PJR ‖22,2≤ c[ 1ε2‖ (I − PJ)R ‖22,2 + ‖ g ‖22,2 +

1ε2‖ Ψ ‖22,2,∼

]. (4.38)

Proof. Let R = FyR, be the Fourier-transform in y of R and ξy ∈ Z the conjugatevariable to y. R satisfies the equation

ivyξyR+ε2

δ2C2Fy

σ(y)vx(vx

∂R

∂vy− vy

∂R

∂vx)

= ε−1LJR+ H1(R)− vyr(−1)ξy + g

(4.39)with r(v) now denoting the difference between ingoing and outgoing boundary val-ues,

r(v) = R(π, v)−R(−π, v). (4.40)With the notations < R > for the 0-Fourier term, and R := R− < R >, we shallfirst give an estimate for R. We use the notation

Z = ε−1LJR+ H1(R)− ε2

δ2C2Fy

σ(y)vx(vx

∂R

∂vy− vy

∂R

∂vx)

+g − (−1)ξyvyr,

Z = ε−1LJR+ H1(R) + g − (−1)ξyvyr,

Z ′ = ε−1LJR+ H1(R) + g, U = (iξyvy)−1.


Let h be the indicatrix function of the set v ∈ R3 |vy| < α, for some positive αto be chosen later.

Let ζs(v) = (1+ | v |)s. For ξy 6= 0

‖ P (hR(ξy, · ) ‖ ≤ c4∑j=0

∣∣∣∣∫R3dvh(v)R(ξy, v)M(v)χj(v)

∣∣∣∣≤ c ‖ ζ−shR ‖

4∑j=0

‖ hζsχj ‖≤ c√α ‖ ζ−shR ‖ .

We use this estimate with the following choice of α, α =‖ ζ−sR ‖−12 ‖ ζ−sZ ′ ‖. We

also introduce an indicatrix function h1 with α = δ1. We fix δ1 so that cδ1 << 1.Then we find from the above estimate that the P -part of the right-hand side, ‖P (h1R) ‖, can be absorbed by ‖ P (h1R) ‖ in the left-hand side. The estimates holdin the same way when h1 is suitably smoothed around

√δ1|ξ|. For the remaining

hchc1R = (1− h)(1− h)R, we shall use that R = −U Z. Then

‖ P (hc1hcR(ξy, · )) ‖2

≤ c ‖ ζs+2hc1hcU ‖2‖ ζ−sZ ′ ‖2 +

‖ P (hc1hcvyr) ‖2

δ1|ξy|2+

ε2

δ2C2|Θ|

≤ c

| ξy |2| α |‖ ζ−sZ ′ ‖2 +


δ1|ξy|2+

ε2

δ2C2|Θ|,

with

Θ =4∑j=0

∫χjh

c1hcUFy

σ(y)vx(vx

∂R

∂vy− vy

∂R

∂vx)Mdv

(∫χjh

c1hc(R− UZ)Mdv

)∗.

We replace α by ‖ ζ−sR ‖−1‖ ζ−sZ ′ ‖ in the denominator. That gives

‖ PR(ξy, · ) ‖2 ≤ c(‖ ζ−sR ‖‖ ζ−sZ ′ ‖ +‖ P (hc1h

cvyr) ‖2

δ1|ξy|2

+δ1 ‖ ζ−s(I − P )R ‖2) +ε2

δ2C2|Θ|.

Hence,

‖ PR(ξy, · ) ‖2≤ c(

(‖ PR ‖ + ‖ ζ−s(I − P )R ‖)2 ‖ ζ−sZ ′ ‖

+‖ P (hc1h

cvyr) ‖2

δ1|ξy|2+ δ1 ‖ ζ−s(I − P )R ‖2

)+

ε2

δ2C2|Θ|.

Consequently,

‖ PR(ξy, · ) ‖2≤ c(‖ ζ−sZ ′ ‖2 +


δ1|ξy|2+ ‖ ζ−s(I − P )R ‖ ‖ ζ−sZ ′ ‖

+ ‖ ζ−s(I − P )R ‖2)

+ε2

δ2C2|Θ|.

We next discuss the term ε2

δ2C2 |Θ|. The first integral can be bounded by an integralof a product of M , 1 + |ξy|, a polynomial in v, | R | and U or U2. Since, by our


choice of δ, we have ε2

δ2C2 = εγ2C2 , this term is bounded by εc ‖ R ‖2. And so,

‖ PR(ξy, · ) ‖2≤ c(‖ ζ−sZ ′ ‖2 +


δ1|ξy|2+ ‖ (I − P )R ‖2

).

Therefore for ξy 6= 0,∫|PR|2(ξy, v)Mdv ≤ c

( 1ε2‖ ζ−s(v)LJR(ξy, ·) ‖22 + ‖ (I − P )R(ξy, ·) ‖2

+‖ P (hc1h

cvyr) ‖2

δ1|ξy|2+ ‖ ν− 1

2 g(ξy, ·) ‖2). (4.41)

The error from evaluating P instead of PJ is of order δ. We remind that thezero Fourier mode of PR is zero by definition. Hence, taking ε small enough andsumming over all 0 6= ξy ∈ Z, implies by Parseval and the spectral estimate on LJ ,that ∫

(PJR)2(y, v)Mdvdy ≤ c( 1ε2

∫ν((I − PJ)R)2(y, v)Mdvdy

+‖ PJ(hc1h

cvyr) ‖2

δ1+∫ν−1g2(y, v)Mdvdy + δ2 ‖ R ‖2

). (4.42)

The r-term can be expressed from (4.39) at ξy = 0

vyr(v) =1εLJR(0, v)− ε2

δ2C2Fy

σ(y)vx(vx

∂R

∂vy− vy

∂R

∂vx) ∣∣∣

ξy=0(4.43)

+g(0, v) + H1(R)(0, v).

Inserting this into (4.42) results in∫(PJR)2(y, v)Mdvdy ≤ c

( 1ε2

∫ν((I − PJ)R)2(y, v)Mdvdy

+∫ν−1g2(y, v)Mdvdy + (δ2 + ε2) ‖ R ‖22,2

).

We are left with the Fourier component PJ R(ξy) for ξy = 0. We estimate separatelythe (I−P )-component and the P -component of PJ R(0, ·). For (I−P )PJ R(0, ·) weobtain

‖ (I − P )PJ R(0, ·) ‖≤ c(ε+ δ)(‖ (I − PJ)R ‖2,2 + ‖ PJR ‖2,2).

For PPJ R it is enough to consider PR, since they differ only by terms of order(ε + δ). We have PR(0, v) =

∑4`=0 χ`(v)

∫dvMψ`

∫dyR(y, v). We discuss each

moment separately.Given two functions h(v) and f( · , v) we use the notation fh( · ) :=

∫dvh(v)f(·, v).

In particular, for h = χj , j = 0, . . . , 4, we also use the notation fj .

• vy-moment:

Multiply (4.16) by k(y)M and integrate over velocity. Now integrate over [−π, y]and v ∈ R3. Since

∫vyM(v)R(±π, v)dv = 0, we have from (4.16) Rvy

(0) =∫vyM(v)R(y, v)dvdy = 0.

• vx-moment:


We estimate the moment Rvxv2y(0) and then use that

Rvx = Rvxv2y− R⊥vxv2y

,

with R⊥ = (1− P )R.Indeed,

∫MPRvxv

2ydv =

∫MRvxdv, because

∫Mv2

xv2ydv = 1. Multiply equation

(4.33) by Mvxvy, integrate over [y, π] and v ∈ R3 and then integrate over y ∈[−π, π]. We get

−∫ π

−πdy

∫dvvxv

2yMR(y, v)dv + 2π

∫dvvxv

2yMR(π, v)dv (4.44)

≤ c[

ε

γC2‖ R ‖2,2 +

1ε‖ (I − PJ)R ‖2,2 + ‖ g ‖2,2

].

In the boundary term the integral over vy < 0 is easy because the boundary con-dition in π is given in terms of Ψ. To control the outgoing (vy > 0) part, wemultiply again equation (4.33) by 2πMvxvy, integrate this time over [−π, π] andv ∈ R3, vy > 0. We get

2π∫vy>0

dvvxv2yMR(π, v)dv − 2π

∫vy>0

dvvxv2yMR(−π, v)dv (4.45)

≤ c[

ε

γC2‖ R ‖2,2 +

1ε‖ (I − PJ)R ‖2,2 + ‖ g ‖2,2

].

To control the second term in l.h.s. of (4.45), we use the boundary condition in −πand notice that the integral is zero. Finally, we replace the bound of the outgoingpart given by equation (4.45) in (4.44). The result is

|Rvxv2y(0)|2 =

∣∣∣∣∫ π

−πdy

∫vxv

2yMR(y, v)dv

∣∣∣∣2 ≤ c[ ε2

γ2C4‖ R ‖22,2 (4.46)

+1ε2‖ (I − PJ)R ‖22,2 + ‖ g ‖22,2 +

1ε2‖ Ψ ‖22∼]

and the same estimate holds for |Rvx(0)|2.• vz-moment:

we get the same estimate for the vz-moment and the proof is analogous, the onlydifference being that we start by multiplying (4.33) by Mvyvz.• χ4-moment:

We recall χ4 = |v|2−3√6

and we denote R4(0) =∫ π−π ψ4MRdvdy. We notice that

Rv2yA(0) =1√6R4(0)

∫v2yv

2AMdv + R⊥v2yA(0), (4.47)

where A is the non-hydrodynamic solution to

L(vyA) = vy(v2 − 5). (4.48)

To control Rv2yA(0) we multiply (4.33) by Mv2zA and proceed as before: first, we

consider the integral∫ π−π dy

∫ πydy′∫

R3 dv, then we study 2π∫ π−π dy

∫vy>0

dv and takethe difference. Now, the analogous of the second term in (4.45) is

2π∫vy>0

dvv2yAMR(−π, v)dv.


We use the boundary condition in −π and observe that∫v2yAMdv = 0 by orthog-

onality and since the integral is even in vy, also∫vy>0

v2yAMdv = 0.

• χ0-moment:

Since∫dvv2

yRMdv = R0 +2R4√

6+∫dvv2

yR⊥Mdv, and we already have estimated

R4(0), it is enough to estimate the moment Rv2y (0). To this end, multiply (4.33)by vyM and integrate over [y, π] × R3. Since vy is in the null of L, we do notget contribution from the L and H1 terms on the right hand side. Moreover, byintegration by parts, there is no contribution depending on PR in the force term.We have

−∫dvv2

yMR(y, v)dv + 2π∫vy<0

dvv2yMR(π, v)dv + 2π

∫vy>0

dvv2yMR(π, v)dv

≤ c( ε

γC2‖ (I − P )R ‖2,2 + ‖ g ‖2,2).

The second term depends on the ingoing flow in π, which is given in terms of Ψ. Tocontrol the third term, we will as before cancel it by considering a new equation.Multiply (4.33) by 2πvyM and integrate over [−π, π]× v ∈ R3 : vy > 0.

2π∫vy>0

dvv2yMR(π, v)dv − 2π

∫vy>0

dvv2yMR(−π, v)dv

≤ c(

ε

γC2‖ R ‖2,2 + ‖ g ‖2,2 +

1ε‖ (I − PJ)R ‖2,2

).

We get

−∫dvv2

yMR(y, v)dv + 2π∫vy>0

v2yMR(−π, v)dv

≤ c(

ε

γC2‖ R ‖2 + ‖ g ‖2 +

1ε‖ (I − PJ)R ‖2 +

1ε‖ Ψ ‖2∼

).

To estimate the second term, we cannot use symmetry arguments as before. Weinstead use Schwartz inequality∫

vy>0

v2yM |R|(−π, v)dv ≤

[∫vy>0

v3yM

]1/2 [∫vy>0

vyMR2(−π, v)dv

]1/2

≤ c ‖ γ−R ‖2 .We now can use (4.32) to estimate ‖ γ−R ‖2 and get

|Rv2y |2 ≤ c[η ‖ PJR ‖22,2 +

1ηε2‖ (I − PJ)R ‖22,2 + ‖ g ‖22,2 +

1ε2‖ Ψ ‖22,2,∼

].

Collecting all the moment estimates for ξy = 0, we get

‖ PJ R(0) ‖22≤ c[η ‖ PJR ‖22,2,+

1ηε2‖ (I − PJ)R ‖22,2 + ‖ g ‖22,2 +

1ε2‖ Ψ ‖22,2,∼

]and this concludes the proof of Lemma 4.4.

Combining the two steps we can prove


Theorem 4.5. Assume δ2 = εγ. Then, for γ small enough, if∫Mgdv = 0, the

solution of (4.33) satisfies

‖ PJR ‖22,2≤ c(‖ (I − PJ)g ‖22,2 +

1ε2‖ PJg ‖22,2 +

1ε4‖ Ψ ‖22,2,∼

)(4.49)

‖ ν 12 (I − PJ)R ‖22,2≤ c ‖ PJg ‖22,2 +

1ε2η‖ Ψ ‖22,2,∼ +ε2 ‖ (I − PJ)g ‖22,2

Proof. Replace (4.22) in (4.38), and use γ and η small.

The step from L2 to L∞ is done by studying the solution along the characteristicsas [1]. The result is

‖ ν 12R ‖2∞,2≤ c

( 1ε2‖ R ‖22,2 +ε2 ‖ ν− 1

2 g ‖2∞,2 +1ε2‖ Ψ ‖22,2,∼

).

Then, by Theorem 4.5,

‖ ν 12R ‖2∞,2≤ c

( 1ε2‖ ν− 1

2 (I − PJ)g ‖22,2 +1ε4‖ PJg ‖22,2 +ε2 ‖ ν− 1

2 g ‖2∞,2

+1ε6‖ Ψ ‖22,2,∼

). (4.50)

With the a priori estimates provided by Theorem 4.5, we are now in the positionof proving the existence theorem for the remainder equation (4.16-4.18), by aniteration procedure.

Theorem 4.6. There exists a solution in L2([−π, π]× R3;Mdvdy) to the problem(4.10-4.13) and is unique.

Proof. The remainder term R will be obtained as the limit of the approximatingsequence Rn, where R0 = 0 and

vy∂Rn+1

∂y+

ε2

δ2C2µ(y)vx(vx

∂Rn+1

∂vy− vy

∂Rn+1

∂vx) =

1ε

[LJRn+1 +H1Rn+1]

+ J(Rn, Rn) + a, (4.51)

Rn+1(−π, v) =M−M

∫−wy>0

(Rn+1(−π,w) +1ε

Ψ(−π,w))|wy|Mdw,

−1ε

Ψ(−π,w), vy > 0, (4.52)

Rn+1(π, v) =M+

M

∫wy>0

1ε

Ψ(π,w)|wy|Mdw − 1ε

Ψ(π, v), vy < 0.

Here a is of order ε4 and it holds that Rn+1 satisfies also the boundary conditions(4.18). The function R1 is solution to (4.33-4.35) with g = a. Then, by using the apriori estimates of Theorem 4.5 and (4.50), together with the exponential decreaseof Ψ, we obtain, for some constant c1,

‖ ν 12R1 ‖∞,2≤ c1ε2, ‖ ν 1

2R1 ‖2,2≤ c1ε3.


By induction for ε sufficiently small,

‖ ν 12Rj ‖∞,2≤ 2c1ε2, j ≤ n+ 1,

‖ ν 12 (Rn+1 −Rn) ‖2,2≤ c2ε

32 ‖ ν 1

2 (Rn −Rn−1) ‖2,2, n ≥ 1,

for some constant c2. Namely,

vy∂Rn+1

∂y+

ε2

δ2C2µ(y)vx(vx

∂(Rn+2 −Rn+1)∂vy

− vy∂(Rn+2 −Rn+1)

∂vx))

=1ε2LJ(Rn+2 −Rn+1) +

1εH1(Rn+2 −Rn+1) +

1εGn+1,

(Rn+2 −Rn+1)(−π, v) =M∓M

∫wz≶0

(Rn+2 −Rn+1)(−π,w)|wy|M−dw, vy > 0,

(Rn+2 −Rn+1)(π, v) = 0, vy < 0

Here, Gn+1 = (I − P )Gn+1 = J(Rn+1 +Rn, Rn+1 −Rn). It follows that

‖ ν 12 (Rn+2 −Rn+1) ‖2,2≤ cε−

12 ‖ ν− 1

2Gn+1 ‖2,2≤ cε− 1

2

(‖ ν 1

2Rn+1 ‖∞,2 + ‖ ν 12Rn ‖∞,2

)‖ ν 1

2 (Rn+1 −Rn) ‖2,2

≤ c2ε32 ‖ ν 1

2 (Rn+1 −Rn) ‖2,2 .

Consequently,

‖ ν 12Rn+2 ‖2,2 ≤‖ ν 1

2 (Rn+2 −Rn+1) ‖2,2 +...+ ‖ ν 12 (R2 −R1) ‖2,2

+ ‖ ν 12R1 ‖2,2≤ 2c1ε3,

for ε small enough. Similarly, ‖ Rn+2 ‖∞,2≤ 2c1ε2. In particular Rn is a Cauchysequence in L2

M ([−π, π]2 × R3). The existence of a solution R to (4.10) follows.Uniqueness follows along the same path.

Corollary 4.7. There exists an isolated L2-solution F to (1.23) such that

‖M−1[F −Mδ] ‖2,2≤ cε.

Proof of Theorem 1.1. The positivity can be proved as in [3]–[6]. Then, by Corol-lary 2.2 and 4.7 we have, for q = 2,∞

‖M−1[F −M(1, 1, (δU, 0, 0))] ‖q,2≤‖M−1[F −Mδ] ‖q,2+ ‖M−1[Mδ −M(1, 1, (δU, 0, 0))] ‖q,2≤ c(ε+ δ2)

which implies (1.25) by taking into account the relation δ2 = γε.

Acknowledgments. R. E. and R. M. are partially supported by Murst-Prin. Thework of A. N. has been supported by the ANR EGYPT. L. A. and A. N. thank theUniversity of L’Aquila and R. E. and R. M. thank the Chalmers University and theIHES for the kind hospitality.


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